The Early History of Pi
Egypt
In 1856 a Scotsman by the name of Henry Rhind purchased a
papyrus in Egypt. A copy dating to 1,650 B.C., it consisted of a collection
of mathematics problems prepared for a typical mathematics student of the
day.
Problems of algebra, linear systems, mensuration, and mathematical amusements were assembled. There was no formal theory, just the enumerated
problems.
In one problem we find the statement:
.
A square of side 8 has the area of a circle of diameter 9.
From this statement, the “student” was expected to compute the area of any
circle. Fortunately, the concept of proportion was well understood and thus
by scaling any circular area was easily determined.
Effectively, using the fact that the true area of a circle of diameter 9 is
2
/
π
= (4 5)2 ≈ 63 62
r
π
.
.
and the area of a square of side 8 is 64, we easily see that the effective value
for is
13
≈ 3 1605 or 3
81
Most modern histories give this value to be
π
π
.
3
1
6
.
From the Bible:
“Also he made a molten sea of ten cubits from brim to brim, round in
compass, and five cubits the height thereof; and a line of thirty cubits did
compass it round about”.
Old Testament, 1 Kings 7:23
Babylon The Baylonians enjoyed a sophisticated society. Located in the
Mesopotamian River Valley, they created large monuments to their gods and
kings while engaging in commerce throughout the Mediterranian.
To do this they required some considerable mathematical skills. Excellent
records of their whole society, including their mathematics has been preserved
in the tens of thousands of clay tablets.
Mathematically ...
For example, they seemed to be well aware of the Pythagorean theorem
a thousand years before Pythagoras. They developed a method of square
root extraction equivalent to Newton’s method. Generally, they used the
approximation
π = 3.
However, at Susa (about 200-300 miles from Babylon) a new set of tablets
were unearthed in 1936. These tablet were on whole concerned with the
ratios of areas and perimeters of regular polygons to their respective side
lengths. For example, we find
The ratio of the perimeter of a regular inscribed
hexagon to the circumference of the circle in sexagesimal numbers is
0; 57, 36 = 0 +
57
35
+ 2 ≈ .9600
60 60
The correct value of this ratio is
6
2π
and this gives an effective value of π to be
3
1
8
.
Greece
The Greeks, living in an open society free of political and other
social restraints, were the first to develop the axiomatic method and rigor
in their mathematics.
Three problems vexed (and blessed) their mathematical development:
• How to square the circle.
• How to double the cube.
• How to trisect an angle.
What they didn’t know is that the first problem depends on the
very nature of π.
What they didn’t know was that these problems would require more than
two millenia to solve.
In Euclid’s Elements, Book XII, Proposition 2, we find the following result.
(It’s proof is based on the Eudoxus Method of Exhaustion, the Greek version
of taking limits, without formally defining them.)
The ratio of the areas of two circles is as the ratio of the
squares upon their diameters.
From this we know that the Greek mathematicians were well aware that
there must be a mathematical model equivalent to the one we know as
A = π r2
Archimedes
The greatest mathematician of antiquity, Archimedes
(287-212 B.C.), proved two fundamental relations about the perimeters and
areas of inscribed and circumscribed regular polygons. With respect to a
circle of radius r, let
b1 = an inscribed hexagon with perimeter p1 and area a1
B1 = an circumscribed hexagon with perimeter P1 and area A1
Further, let b2 . . . . bn denote the regular inscribed 6 · 2 . . . 6 · 2n polygons,
similarly, B2 . . . Bn for the circumscribed polygons.
The following formulae give the relations between the perimeters and areas
of these 6 · 2n polygons.
2pn Pn
pn + Pn
p
= an An
Pn+1 =
pn+1 =
an+1
An+1 =
p
pn Pn+1
2an+1 An
an+1 + An
Using n-gons up to 96 sides he derives
3
10
1
<π<3
71
7
Following Archemedes, Apollonius (250-175 B.C.), in his book, now lost,
on rapid calculation techniques, Quick Delivery, approximates π as
π ≈ 3.1416
No information on how he derived it is known. Apollonius is considered one
of the last great mathematicians of antiquity. His book, On Conics, did for
conics what Euclid did for plane geometry.
The great astronomer and the father of trigonometry Claudius Ptolemy
(100-178 A.D.) gives
377
π≈
= 3.141666666...
120
although his method of derivation is unknown. His book, commonly known
as Almagest, was studied and considered the source of trigonometry up to
the time of Copernicus (fl. 1500)
Rome
The Romans were warriors, engineers, authors, and rulers. They
were not mathematicians. Even with all the Greek mathematics available,
they computed circular areas using proportion and the statement
A wheel of diameter 4 feet has a perimeter of 12 12 feet.
This gives the approximation
π≈3
1
8
China
Generally, the Chinese, even before
√ 100 B.C., used π ≈th3. They
also used the alternative approximation, π ≈ 10 ≈ 3.16. By the 5 century
more accurate approximations were known. Tsu Ch’ung-Chih gives two sets
of approximations
22
inexact →
7
355
= 3.14159292035
accurate →
113
He also shows
3.1415926 < π < 3.1415927
Note that to eleven decimal places π = 3.14159265359. Results of this accuracy were not achieved in the West until the 15th century.
Hindu
The best approximation to π was given in about 500 A.D. by
Āryabhata using the following description
Add 4 to 100, multiply by 8, and add 62,000.
The result is approximately the circumference of
a circle of which the diameter is 20,000.
Of course this gives
(100 + 4)8 + 62, 000
177
=3
= 3.1416
20, 000
1250
Yet, the Hindu approximation to π was usually
√
10.
Arabic In 833 A.D., the great arab mathematician Al-Khwārizmī, the
father of algebra, uses two approximations for π: 3 17 and
π=
62, 832
= 3.1416
20, 000
In about 1436 Al-Kashi gives
2π ≈ 6.2831853071795865,
sixteen decimal places, which is prodigiously accurate for the time.
The Modern History of Pi
France The French mathematician Francois Vieta (1540-1603), the first
mathematician to us symbolic notation for knowns (consonants) and unknowns (vowels) discovered the first direct formula for the calculation of
π.
π
=
2
q r
1
2
1
2
+
1
2
q
1
2
s
1
1
2
+
1
2
r
1
2
+
1
2
q
1
2
···
This is easily derived beginning with an inscribed square, and successively
doubling the number of sides, getting next the octagon, the 16-gon, .... Using
the notation an for the area of the inscribed n-gon, apply the formula
π
a2n = an sec
n
recursively, beginning with n = 4 (the square). Calculating thusly, Vieta
obtained ten places accuracy.
Germany
Ludolph van Ceulen (1540-1610) computed π accurate to
20 places. Later, he computed π to 31 places. So proud was he of this
accomplishment that he had the digits engraved on his tombstone. At this
time π is sometimes called Ludolph’s constant. (It is not yet called π.)
The known accuracy of π now exceeds any
computational need.
England.
A number of mathematical developments come from England. The first approximation to π came from the clergyman, cryptographer and mathematician John Wallis (1616-1703). Using a complicated
interpolation-like argument based on the quadratures of the functions (1 +
x)n , n = 0, . . ., he produced
2
1 · 3 · 3 · 5 · 5 · 7···
=
π
2 · 2 · 4 · 4 · 6 · 6···
From this Lord William Brouncker (1620?-1684), the first president of
the Royal Society, discovered the formula
4
=1+
π
1
9
2+
2+
25
49
2+
2 + ···
James Gregory (1637-1675) derived the well known (to us) formula
Z
0
x
x5
x7
dt
x3
+
−
+ ···,
=
arctanx
=
x
−
1 + t2
3
5
7
now called Gregory’s formula. Evaluating at x = 1, we get
1 1 1
π
= 1 + − + − ···
4
3 5 7
This series was used by Abraham Sharp (1651-1742) under instructions
from Edmund Halley (1656-1743) (of comet fame) to calculate π to 72 places.
(Note the convergence is very slow. How many terms are required for 10−72
accuracy?)
John Machin (1680-1751), a professor of astronomy at Gresham College
in London obtained 100 places of accuracy for the computation of π using
Gregory’s formula
1
1
π
= 4 arctan − arctan
4
5
239
Even Issac Newton in 1665-1666 calculated π from the formula
√
3 3
1
1
1
+ 24( −
π=
−
+ · · ·)
5
4
12 5 · 2
28 · 27
√
Z 14 p
3 3
x − x2 dx
=
+ 24
4
0
Said Newton:
“I am ashamed to tell you to how many figures I carried these computations,
having not other business at the time.”
In 1874, Machin’s formula was used by William Shanks (1812-1882) to
achieve 707 places of accuracy for π. Because the series is alternating and
almost geometric, the convergence is very, very fast. Only about 510 terms
are required to achieve this accuracy.
Germany.
Perhaps the greatest mathematician that ever lived, Leonhard Euler (1707-1783), was born in Basel and studied under James Bernoulli.
He eventually found a position at St. Petersburg, originally to serve on the
faculty of medicine and physiology.
In 1730, due in part to the death of his colleague and friend, Nicolas Bernoulli,
he found himself in the chair of natural philosophy–not medicine. His contributions to mathematics have been well documented and are prodigious.
He also contributed to the derivation of formulas with π.
In 1779 he determined the formula
1
3
π = 20 arctan − 8 arctan
7
79
from which the computation of π can be made using Gregory’s formula.
However, Euler derived a different formula for the arc tangent as well. He
also derived the formulas
∞
X
π2
1
=
6
n2
n=1
∞
X
π4
1
=
90 n=1 n4
∞
X
π6
1
=
945 n=1 n6
∞
X
π8
1
=
9, 450 n=1 n8
∞
X
π10
1
=
93, 955 n=1 n10
..
.
=
..
.
as well as formulas for the sum of the remaining reciprocal (even) integers
up to to order 26.
Euler coined the modern symbol for pi, π, after John Wallis who first used
√
π
for pi. In 1801 he also used the modern symbols for e and i = −1.
δ
Euler proved the most important relation in all of mathematics:
eiπ + 1 = 0
The Nature of Pi
Johann Lambert (1728-1777) was born in the Alsace region, the son of a
poor tailor. Managing to teach himself elementary mathematics and pursue
his mathematical studies while a tutor for a Swiss family, he eventually was
able to enjoy the company of the day’s greatest mathematicians including
Lagrange and Euler. In 1761 he proved that
if x is rational (and non-zero), then neither ex
nor tan x can be rational.
He proves this using a continued fraction expansion of e given by Euler in
1737.
Note.
At this time there were so many circle squarers that in 1775 the
Paris Academy passed a resolution that no more solutions on the quadrature
of the circle should be examined by its officials. This resolution applied as
well to angle trisections and the doubling of the cube. The conviction was
growing that the solution of squaring the circle was impossible.
It can be shown that by using a straight edge and compass, the only magnitudes that can be constructed from whole magnitudes are numbers involving
square roots, square roots of square roots, etc. For example,
r
q
q
√
√
√
2,
3 + 4,
5 + 6 + 7, etc.
Such numbers are also among the solutions of all polynomial equations with
rational coefficients. The set of all solutions, real and complex, of polynomial
equations with rational coefficients are called the algebraic numbers. All
other numbers are called transcendental.
Transcendental numbers were discovered by Joseph Liouville (1809-1882)
in 1844 with the number
1
1
1
+ 2! + 3! + · · ·
10 10
10
In 1873, Charles Hermite (1822-1901) proved that e was transcendental
by showing that it cannot be the solution of a polynomial equation with
rational coefficients.
Finally, in 1882, C.L.F. Lindemann (1852-1939) proved that π is transcendental by showing that if a, b, c, . . . and m, n, r, . . . are algebraic numbers
the the equation
aem + ben + cer + · · · = 0
has no rational solutions. Thus the equation
eix + 1 = 0
cannot subsist if x is algebraic. Using Euler’s celebrated formula
eiπ + 1 = 0
it follows that π is not algebraic. The circle cannot be squared!
Bill No. 246, 1897. State of Indiana
“Be it enacted by the General Assembly of the State of Indiana: It has been
found that the circular area is to the quadrant of the circumference, as the
area of an equilateral rectangle is to the square on one side.”
“In further proof of the value of the author’s (E.J.Goodman, M.D.) proposed
contribution to education, and offered as a gift to the State of Indiana, is
the fact of his solutions of the trisection of the angle, duplication of the cube
and quadrature of the circle having been already accepted as contributions
to science by the American Mathematical Monthly, the leading exponent of
mathematical thought in this country.”
Passes three readings, Indiana House, 1897.
(via Committee on Swamp Lands)
Passes first reading, Indiana Senate, 1897.
(via Committee on Temperance)
“The case is perfectly simple. If we pass this bill which establishes a new
and correct value of π, the author offer our state without cost the use of this
discovery and its free publication in our school textbooks, while everyone
else must pay him a royalty.”
Professor C.A. Waldo of Purdue intercedes and the bill is tabled.
What we do know!
?
?
?
?
π is irrational.
π is √
transcendental
π + 2 log 2 is transcendental
Several billion digits behave regularly.
Lambert (1761)
Lindemann (1882)
Baker (1966)
Sophisticated Computations
?
?
?
?
?
?
(or what can be done with really fast computers)
Besides the computations already mentioned Machin-like formulae have produced
2037 Digits – Eniac (1949) – (von Neumann, et.al.)
1,000,000 Digits – CDC 7600 (1973) – (Guilloud and Bouyer)
17,000,000 Digits – Symbolics 3670 (1985), (Gosper)
29,000,000 Digits – Cray-2 (1986), (Bailey)
1-4 Billion Digits – (Chudnovskys)
51 Billion Digits – Hitachi’s (1997) – (Kanada)
India
In 1914 the Indian mathematician Srinivasa Ramanujan proved
the following representation of π.
√ ∞
8 X (4n)![1103 + 26390n]
.
1/π =
9801 n=0
(n!)4 3964n
Canada
Even better formulas have been more recently developed. For
example, in 1987, Peter and Jonathan Borwein of the University of Waterloo, formulated a Ramanujan-like formula given below: With
√
a = 212175710912 61 + 1657145277365
and
√
b = 13773980892672 61 + 107578229802750
we have
∞
X
1
(−1)n (6n)![a + bn]
√
= 12
4 (3n)![5280(236674 + 30303 61)][3n+3/2]
π
(n!)
n=0
This formula delivers an impressive 25 digits of accuracy for each value of n
computed. Compute to 99 terms and you will obtain π to 2,500 digits!!!
Iterative Formulas are available also. From the Borweins we also have
y0 =
√
2−1
√
α0 = 6 − 4 2
1
yn+1
1 − (1 − yn4 ) 4
=
1
1 + (1 − yn4 ) 4
2
αn = (1 + yn+1 )4 αn − 22n+3 yn+1 (1 + yn+1 + yn+1
)
It follows that
1
→π
αn
as n → ∞.
With n at 15 you are guaranteed over 2 billion places of accuracy for π.
The Current Record. 20th September 1999
Japan
The current record is held by Dr. Yasumasa KANADA of the
Computer Centre, University of Tokyo Dear folks,
Our latest record was established as the followings;
Declared record: 206,158,430,000 decimal digits
Two independent calculation based on two different algorithms generated
206, 158, 430, 208(= 3 ∗ 23 6) decimal digits of pi and comparison of two generated sequences matched up to 206,158,430,163 decimal digits, e.g., 45 decimal digits difference. Then we are declaring 206,158,430,000 decimal digits
as the new world record.
Optimized Main program run:
Job start : 18th September 1999 19:00:52 (JST)
Job end : 20th September 1999 08:21:56 (JST)
Elapsed time : 37:21:04
Main memory : 865 GB (= 6.758 GB * 128)
Algorithm : Gauss-Legendre algorithm
Optimized Verification program run:
Job start : 26th June 1999 01:22:50 (JST)
Job end : 27th June 1999 23:30:40 (JST)
Elapsed time : 46:07:10
Main memory : 817 GB (= 6.383 GB * 128)
Algorithm : Borwein’s 4-th order convergent algorithm
Some of interesting digits sequences of pi:
01234567890 : from 53,217,681,704-th of pi
01234567890 : from 148,425,641,592-th of pi
01234567891 : from 26,852,899,245-th of pi
01234567891 : from 41,952,536,161-th of pi
01234567891 : from 99,972,955,571-th of pi
01234567891 : from 102,081,851,717-th of pi
01234567891 : from 171,257,652,369-th of pi
432109876543 : from 149,589,314,822-th of pi
543210987654 : from 197,954,994,289-th of pi
98765432109 : from 123,040,860,473-th of pi
98765432109 : from 133,601,569,485-th of pi
98765432109 : from 150,339,161,883-th of pi
98765432109 : from 183,859,550,237-th of pi
09876543210 : from 42,321,758,803-th of pi
09876543210 : from 57,402,068,394-th of pi
09876543210 : from 83,358,197,954-th of pi
10987654321 : from 89,634,825,550-th of pi
10987654321 : from 137,803,268,208-th of pi
10987654321 : from 152,752,201,245-th of pi
27182818284 : from 45,111,908,393-th of pi
Frequency distribution
’0’ : 20000030841; ’3’ :
’1’ : 19999914711; ’4’ :
’2’ : 20000136978; ’5’ :
for pi up to 200,000,000,000 decimal places:
20000069393; ’6’ : 1999988151; ’9’ : 19999869180
19999921691; ’7’ : 19999967594
19999917053; ’8’ : 20000291044; Chi square = 8.0
Programs are consisted of two sets of routines, e.g. calculation routines and
message passing routines. Calculation routines were written by Dr. Daisuke
TAKAHASHI, a Research Associate at our Centre and rather speed sensitive
message passing routines were written by myself. Calculation routines used
were more optimized than these used for the 51.5 billion record establishment.
For establishing this new record, high speed message passing routines were
seriously used for both of programs, e.g. main program and verification
program. CPU used was HITACHI SR8000 at the Information Technology
Center, Computer Centre Division (old Computer Centre,) University of
Tokyo. Full of the total CPU, e.g. 128PE’s (theoretical peak processing
speed for the single PE is eight billion floating operations per second. One
trillion floating point operations per second for all PE’s), were definitely used
as single job and parallel processing for both of programs run.
URL:http://www.lupi.ch/PiSites/Pi-Rekord.html
Yasumasa KANADA
Computer Centre, University of Tokyo
Bunkyo-ku Yayoi 2-11-16
Tokyo 113 Japan
Fax : +81-3-3814-7231 (office)
E-mail: kanada@pi.cc.u-tokyo.ac.jp
Is that it?
I believe so!